Questions & Answers

Question

Answers

A. $\sin x$ +C

B. $\cos x$ +C

C. C

D. None of these

Answer
Verified

From the integration it is clear that it is a product of two functions of x. Therefore, integration by parts should be used to evaluate the integral.

Let us assume x as the first function and as the second function.

\[I = x\int {\sin xdx} - \int {\left[ {\left( {\dfrac{{d\left( x \right)}}{{dx}}} \right)\int {\sin xdx} } \right]dx} ......(2)\]

The integration of $\int {\sin xdx = - co} sx$ and the differentiation of $x = 1$ , put the value in equation (2)

$I = - x\cos x + \int {\cos xdx} ......(3)$

The integration of \[\int {\cos xdx = \sin } x\] put the value in equation (3)

$I = - x\cos x + \sin x + C......(4)$

Where, C is a constant and known as constant of integration.

Comparing equation (4) and (1), it is clear that

The value of $\alpha = \sin x + C$

Hence, the correct option is (A).

For instance in $\int {x\cos x} $ x is the first function and is the second function because x is the algebraic function and cosine function is the trigonometric function. So the preference for first function is given to x.

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